Step 1: Understanding Equivalence Relations

An equivalence relation on a set A is a relation that is reflexive, symmetric, and transitive. There is a one-to-one correspondence between equivalence relations on a set and the partitions of that set.

Step 2: Identifying Partitions of Set A

The set A = {1, 2, 3} has 3 elements. We need to find the number of ways to partition this set, which is given by the Bell number B₃.

Step 3: Listing the Partitions

The possible partitions are:

• 1 partition of size 3: {{1, 2, 3}}
• 3 partitions of size 2 and 1: {{1, 2}, {3}}, {{1, 3}, {2}}, {{2, 3}, {1}}
• 1 partition of size 1, 1, and 1: {{1}, {2}, {3}}

Summing these up, we get 1 + 3 + 1 = 5.